A new breed of health insurers created under the Affordable Care Act — representing one of the government’s most innovative attempts in decades to foster better coverage — is on shaky financial ground in many of the 23 states where the plans began.

The nonprofit health plans were envisioned as a consumer-friendly counterweight to for-profit insurers, a way to provide more competition, greater consumer choice and better coverage in markets typically dominated by big commercial carriers. The government allocated billions of dollars in loans for them.